My understanding of the Fourier transform (DFT,FFT)

I am not a major in communication, close Professional + hard to finally let me understand the communication theory of some knowledge, I firmly believe that I do not move. Read a few days of the book, summed up, in modern communication, Fourier transform is a very important component. Modern communication is basically digital communications, There is a lot of understanding of digital signal processing, and learning signal processing before, is to learn the signal and system, read the book only to know the matter, so the non-professional people learn the road is often bent forward, but this bending process will give people a deeper understanding of knowledge.

    especially with the development of communication technology, more mathematics is used in communication, the use of such mathematical knowledge makes it necessary to use complex hardware to achieve the function of the software is easily resolved, The advantage of this is that the proportion of hardware in the design of the product will be smaller, the cost will naturally be reduced. In the 4G era, communication protocols are used to calculate the mathematical aspects of communication, and FFT is becoming more and more important in 4G communication, if the FFT is not understood or understood, Research and development of 4G related products will become very difficult to pursue.

    in the school Fourier transform, a variety of Fourier transforms let me often confuse them, make me dizzy. Ask a colleague who is learning to communicate some knowledge, and later found out that the elder brother never said to point, that is, those who are the most important, He is not willing to say anything that is most confusing. But this does not hinder me, because I am not afraid of this situation, I was in this environment grew up, as long as I want to learn things, I have never been stumped, overcome too many difficulties let me have confidence in myself. Later summed up a generalist found that In fact, that thing just know the essentials, will eventually bypass a lot of detours.

    in communication, our Fourier transform time is a periodic discrete signal in the time domain to the frequency domain of the periodic discrete signals between the transformation, so that is the digital communication, if the transformation has a continuous analog, it is not digital communication. So, It is important to note this point in the use of learning. With this direction, you should know what you should remember and what kind of Fourier transform you should learn.
    

Learned things for a few days to forget, the first few days to see, and now began to become blurred, it seems to learn something or to often review is. The Fourier transform we used to calculate in digital communication is generally referred to as discrete signals in both time and frequency domain. Here we need to make clear what the signal of the periodic signal, aperiodic signal, continuous signal, discrete signal is, and it's good to understand the DFT.

First, we first know the convention that in communication, the variables in the time domain are generally represented by lowercase letters, whereas variables in the frequency domain are generally represented by uppercase letters.

Continuous signal, it should not be explained, that is, the continuous signal in the time domain refers to the amplitude in the time domain of the continuous change of the signal, in the form of X (t) expression, empathy frequency domain is a continuous signal is the amplitude in the frequency domain with the frequency of continuous changes in the signal, generally with similar x (JW) And so on. The non-sequential signal is self-evident means that there is intermittent signal, discontinuous signal, discrete signal, in the digital communication generally refers to similar pulses and other signals.

The next step is the periodic signal and the non-periodic signal. Continuous periodic signals are well understood, such as sine waves, sawtooth waves, and so on. But for discrete signals, it's important to pay attention. In the case of discrete signals, periodic signals generally refer to pulses of equal intervals. and a single pulse or no periodic pulse is a non-periodic signal.

When we learn the signal and system, for the Fourier transform, the book will talk about the changes in various situations, people are very confused, memory is difficult to remember, make a confused. After knowing the above several signals, we can easily remember the Fourier transform in various cases. Let's talk about it next time. Some of the signal types in the Fourier transform are described earlier, and understanding these types is important for us to understand the Fourier transform. So, each type of its transformation, although the truth can be interconnected, but the actual method or formula is not the same,

We should pay attention to another thing, the Fourier transform is a time-and frequency-domain transformation, whereas the Fourier inverse transform is the transformation of the frequency domain and the time domain. We must not confuse this direction.

1. Fourier transform and inverse transformation of a continuous non-periodic time signal:

From the above transformation, we can see that for the non-periodic continuous time signal Fourier transform, its frequency domain is also the non-periodic continuous frequency function.

2. Fourier transform and inverse transformation of the time signal for a continuous period:

From this equation we can see that the time is continuous, but the frequency domain is a discrete quantity. The positive transformation is what we call the Fourier series.

3. Fourier transform and inverse transformation of discrete non-periodic time signals:

From the formula of positive transformation, it can be seen that the discrete non-periodic signal in time is a continuous signal in the frequency domain. Of course, in the formula we can not clearly see the frequency domain of the periodicity. But the reality is that.

4. Fourier transform and inverse transformation of a discrete period of time signal:

We can see that time-discrete periodic signals are also discrete and periodic signals in the frequency domain. This is the DFT that we use to perform calculations in our communications.

In fact, the Fourier transform is also in these four cases, then from these four cases we can summarize their laws, namely:

Non-cyclical <---> continuous

Cycle <---> discrete

This rule is symmetrical for signals in the time and frequency domains. For example, if the time domain is non-cyclical, the signal on the frequency domain must be continuous, and if the signal on the time domain is periodic, the signal on the frequency domain is definitely discrete. and vice versa.

The Fourier transform has different transformation methods under different signal forms, the previous one I talked about the Fourier transform of several signal forms and the relationship between them. I don't care much about the first three types of Fourier transforms. Digital signal processing is basically the final form of processing, That is, in the time and frequency domain are discrete periodic signals of the Fourier transform, the communication used in the operation is also through this form of Fourier transform. Let's review:

The previous formula is to convert the time domain signal to the signal in the frequency domain, that is, we often say that the discrete signal of the Fourier transform, that is, the DFT, the following formula is to convert the frequency domain signal to the signal at the last domain, that is, we often say that the discrete signal of the inverse Fourier transform, namely IDFT.

In the communication process, many of our actual signals are analog signals, such as music, singing. Analog signals to be processed in a computer must first be converted to a digital signal that can be used by the computer. Take the sound, the frequency of the sound from dozens of Hz to 20KHZ, We need to sample the analog signal before turning it into a digital signal. According to the sampling theorem, the sampling rate must be greater than twice times the maximum frequency of the signal to be able to recover the previous analog signal after sampling. Now the sound card is generally used 44KHZ sampling rate, 44khz>20khz*2, That's actually what this means.

The analog signal is sampled to form a string of numbers that are stored in the computer using an array, that is, the X (n) in the formula above, and N is the n sampling point. Then after the conversion, X (k) is used to store the amplitude at different frequency points. K corresponds to the number of samples on the frequency domain.

We use the DFT formula to illustrate that X (k) is used to store amplitude values on different frequencies after the transformation. The frequency point K can be taken from 0 onwards, but the interval of each frequency point is the resolution of the frequency ferr is determined by the sampling rate srate and the number of sampling points taken on the time domain N, that is, the following relationship:

ferr=srate/n

For example, if the sample rate is 44KHZ, that is, 1 seconds to sample 44K sampling points, in practice, if we only take the successive 22K sample points for analysis, then n==22000, then X (k) Store the difference is the frequency of 0hz,2hz,4hz,8hz ..... The magnitude of the point, that is, the resolution of the frequency of X (k) is 44khz/22k=2hz. Next, we can convert the sampled data from the time domain into the data in the frequency domain by the DFT and IDFT formulas, and convert the data from the frequency domain to the data in the time domain.

However, when we calculate, we will find such a problem. When there are too many sampling points, that is, when n is particularly large, the number is calculated for each point on the other field, and its calculation is very large. So we need a fast algorithm to transform, We are talking about FFT and Ifft fast Fourier transform and fast Fourier inverse transform.

We put forward the formula (1):

Equation (1), which can be unfolded by Euler formula:

Then the formula (1) can be expressed in the form of trigonometric functions, we do not have to study the operation of the process, we will eventually get a conclusion, to be transformed, the operation of the DFT, its addition and multiplication of the number of times are approximately equal to the square of n is proportional, but n relatively large time, such as n=1024 is, Complex multiplication of 100多万次 is required, so that the calculation of the computer becomes very slow, which is what we do not want when dealing with transformations.